Warped Product Semi-slant Submanifolds Research Articles

Geometry of Warped Product CR and Semi-Slant Submanifolds in Quasi-Para-Sasakian Manifolds

In the present paper we study the existence or non-existence of warped product semi-slant submanifolds in quasi-para-Sasakian manifolds and prove that there are no proper warped product semi-slant submanifolds in a quasi-para-Sasakian manifold such that totally geodesic and totally umbilical submanifolds of warped product are proper semi-slant and invariant (or anti-invariant), respectively.

Geometric Mechanics on Warped Product Semi-Slant Submanifold of Generalized Complex Space Forms

In this study, we develop a general inequality for warped product semi-slant submanifolds of type M n = N T n 1 × f N ϑ n 2 in a nearly Kaehler manifold and generalized complex space forms using the Gauss equation instead of the Codazzi equation. There are several applications that can be developed from this. It is also described how to classify warped product semi-slant submanifolds that satisfy the equality cases of inequalities (determined using boundary conditions). Several results for connected, compact warped product semi-slant submanifolds of nearly Kaehler manifolds are obtained, and they are derived in the context of the Hamiltonian, Dirichlet energy function, gradient Ricci curvature, and nonzero eigenvalue of the Laplacian of the warping functions.

Warped products semi-slant and pointwise semi-slant submanifolds on Kaehler manifolds

The present paper is to study estimation of inequalities for warped product semi-slant submanifolds of Kaehler manifolds. Next we obtain several fundamental results of pointwise semi-slant submanifolds and their warped products in Kaehler manifolds.

A class of warped product submanifolds of Kenmotsu manifolds

In 2018 Naghi et al. studied warped product skew CR-submanifold of the form M = M1×f M⊥ of a Kenmotsu manifold M¯ throughout the paper , where M1 = MT ×Mθ and MT , M⊥, Mθ represents invariant, antiinvariant, proper slant submanifold of M¯ . Next, in 2019 Hui et al. studied another class of warped product skew CR-submanifold of the form M = M2 ×f MT of M¯ , where M2 = M⊥ × Mθ . The present paper deals with the study of a class of warped product submanifold of the form M = M3 ×f Mθ of M¯ , where M3 = MT × M⊥ and MT , M⊥, Mθ represents invariant, antiinvariant and proper pointwise slant submanifold of M¯ . A characterization is given on the existence of such warped product submanifolds which generalizes the characterization of warped product contact CR-submanifolds of the form M⊥ ×f MT , studied by Uddin et al. in 2017 and also generalizes the characterization of warped product semi-slant submanifolds of the form MT ×f Mθ , studied by Uddin in the same year. Beside that some inequalities on the squared norm of the second fundamental form are obtained which are also generalizations of the inequalities obtained in the just above two mentioned papers respectively.

Chen–Ricci inequality for warped products in Kenmotsu space forms and its applications

Recently, B.-Y. Chen estabished a general relationship between Ricci curvature and the mean curvature vector of a submanifold in Riemannian manifolds. Later, the same inequality was derived for other structures, but not for warped products. In this paper, we derive Chen-Ricci inequality for warped product semi-slant submanifolds in Kenmotsu space forms. Many applications are given.

Warped product semi-slant submanifolds in locally conformal Kaehler manifolds II

In 1994 N.~Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of $CR$- and slant-submanifolds, \cite{MR0353212}, \cite{MR760392}.
 In particular, he considered this submanifold in Kaehlerian manifolds, \cite{MR1328947}.
 Then, in 2007, V.~A.~Khan and M.~A.~Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, \cite{MR2364904}.
 Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and we gave a necessary and sufficient conditions of the two distributions (holomorphic and slant) be integrable.
 Moreover, we considered these submanifolds in a locally conformal Kaehler space form.
 In the last paper, we defined $2$-kind warped product semi-slant submanifolds in almost hermitian manifolds and studied the first kind submanifold in a locally conformal Kaehler manifold.
 Using Gauss equation, we derived some properties of this submanifold in an locally conformal Kaehler space form, \cite{MR2077697}, \cite{MR3728534}.
 In this paper, we consider same submanifold with the parallel second fundamental form in a locally conformal Kaehler space form.
 Using Codazzi equation, we partially determine the tensor field $P$ which defined in~\eqref{1.3}, see Theorem~\ref{th4.1}.
 Finally, we show that, in the first type warped product semi-slant submanifold in a locally conformal space form, if it is normally flat, then the shape operators $A$
 satisfy some special equations, see Theorem~\ref{th5.2}.

Characterization theorems on warped product semi-slant submanifolds in Kenmotsu manifolds

In this paper, we deal with the study of warped product semi-slant submanifolds isometrically immersed into a Kenmotsu manifold. We prove two characterization theorems for a warped product semi-slant submanifold in Kenmotsu manifolds in terms of the tensor fields.

Geometric inequality of warped product semi-slant submanifolds of locally product Riemannian manifolds

AbstractIn the present article, we derive an inequality in terms of slant immersions and well define warping function for the squared norm of second fundamental form for warped product semi-slant s...

An inequality for warped product submanifolds of a locally product Riemannian manifold

Recently, Sahin studied the warped product semi-slant submanifolds of locally product Riemannian manifolds. In this paper, we obtain some geometric properties of such submanifolds with an example. Also, we establish a sharp relationship between the squared norm of the second fundamental form and the warping function in terms of the slant angle. The equality case is also considered.

On warped product semi-slant submanifolds of nearly Trans-Sasakian manifolds

In this paper, we study warped product semi-slant submanifold of type M = NT xf N? with slant fiber, isometrically immersed in a nearly Trans-Sasakian manifold by finding necessary and sufficient conditions in terms of Weingarten map. A characterization theorem is proved as main result.

Warped product semi-slant submanifolds in locally conformal Kaehler manifolds

In 1994, in [13], N. Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of CR- and slant-submanifolds. In particular, he considered this submanifold in Kaehlerian manifolds, [13]. Then, in 2007, V. A. Khan and M. A. Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, [11]. Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and gave a necessary and sufficient conditions for two distributions (holomorphic and slant) to be integrable. Moreover, we considered these submanifolds in a locally conformal Kaehler space form, [4]. In this paper, we define 2-kind warped product semi-slant submanifolds in a locally conformal Kaehler manifold and consider some properties of these submanifolds.

Another class of warped product submanifolds of Kenmotsu manifolds

In this paper, we study semi-slant submanifolds and warped product semi-slant submanifolds of Kenmotsu manifolds, which have not been considered in earlier studies. A characterization and a general sharp inequality for the lower bound of the squared norm of second fundamental form of such immersions are obtained. The necessary condition is obtained for the warped products to satisfy the equality case of the derived inequality. We also give non-trivial examples of such submanifolds.

Geometry of warped product semi-slant submanifolds of Kenmotsu manifolds

In this paper, we study semi-slant submanifolds and their warped products in Kenmotsu manifolds. The existence of such warped products in Kenmotsu manifolds is shown by an example and a characterization. A sharp relation is obtained as a lower bound of the squared norm of second fundamental form in terms of the warping function and the slant angle. The equality case is also considered in this paper. Finally, we provide some applications of our derived results.

Estimation of inequalities for warped product semi-slant submanifolds of Kenmotsu space forms

In this paper, we construct the geometric inequalities for the squared norm of the mean curvature and warping functions of warped product semi-slant submanifolds in Kenmotsu space forms. The equality cases are also discussed.

Geometry of Warped Product Semi-Slant Submanifolds of Nearly Kaehler Manifolds

Non-existence of warped product semi-slant submanifolds of Kaehler manifolds was proved in Sahin (Geom Dedic 117:195–202, 2006), it is interesting to find their existence in a more general setting, e.g., nearly Kaehler manifolds. In this paper, we obtain a necessary and sufficient condition for a semi-slant submanifold of a nearly Kaehler manifold to be a locally warped product. Also, we establish an inequality for the squared norm of the second fundamental form in terms of the warping function and the slant angle. Furthermore, the equality case of the statement is also considered.

Generalized inequalities on warped product submanifolds in nearly trans-Sasakian manifolds

AbstractIn this paper, we study warped product submanifolds of nearly trans-Sasakian manifolds. The non-existence of warped product semi-slant submanifolds of typeNθ×fNTis shown, whereas some characterization and new geometric obstructions are obtained for the warped products of typeNT×fNθ. We establish two general inequalities for the squared norm of the second fundamental form. The first inequality generalizes derived inequalities for some contact metric manifolds (Kadriet al.in J. Korean Math. Soc. 42:1101-1110, 2005; Munteanu in Publ. Math. (Debr.) 66:75-120, 2005; Mustafaet al.in Taiwan. J. Math. 17:1473-1486, 2013; Uddin and Khan in J. Inequal. Appl. 2012:304, 2012), while by a new technique, the second inequality is constructed to express the relation between extrinsic invariant (second fundamental form) and intrinsic invariant (scalar curvatures). The equality cases are also discussed.MSC:53C40, 53C42, 53C15.

Warped product submanifolds of a Kenmotsu manifold

In the present paper, we study warped product semi-slant submanifolds of a Kenmotsu manifold. We obtain some results on the existence of such type warped product submanifolds of a Kenmotsu manifold with an example.

Warped product semi-slant submanifolds in Kenmotsu manifolds

In this paper, we research the existence or non-existence of warped product semi-slant submanifolds in Kenmotsu manifolds. Consequently, we see that there are no proper warped product semi-slant submanifolds in Kenmotsu manifolds such that totally geodesic and totally umbilical submanifolds of warped product are proper semi-slant and invariant (or anti-invariant), respectively.

Warped product semi-slant submanifolds of a locally product Riemannian manifold

In [20], it was shown that there are no warped product semi-slant submanifolds of Kaehler manifolds. In this paper, we show that there exists a class of non-trivial warped product semi-slant submanifolds of a locally product Riemannian manifold contrary to Kaehlerian case. Then, we give a characterization, an example and investigate the geometry of warped product semi-slant submanifolds in a locally product Riemannian manifold.

Warped product submanifolds of Kaehler manifolds with a slant factor

Recently, we showed that there exist no warped product semi-slant submanifolds in Kaehler manifolds. On the other hand, Carriazo introduced anti-slant submanifolds as a particular class of bi-slant submanifolds. In this paper, we study such submanifolds i